Parametric instability of a spinning pretwisted beam under periodic axial force

Abstract

This paper addresses the parametric instability of a cantilever pretwisted beam rotating around its longitudinal axis under a time-dependent conservative end axial force which contains a steady-state part and a small periodically fluctuating component. This structural element can be used to model fluted cutting tools such as the twist drill bit and the end milling cutter, etc. Using the Euler—Bernoulli beam theory and Hamilton's principle, the present study derives the equation of motion which governs the lateral vibration of a spinning pretwisted beam. Rotary inertia, structural viscous damping and conservative end axial force are included. The Galerkin method is then applied to obtain the associated finite element equations of motion. Due to the existence of the Coriolis force, the resulting finite element equations of motion are transformed into a set of first-order simultaneous differential equations by a special modal analysis procedure. This set of simultaneous differential equations is solved by the method of multiple scales, yielding the system response and expressions for the boundaries of the unstable regions. Numerical results are presented to demonstrate the effects of pretwist angle, spinning speed and steady-state part of the end axial force on the parametric instability regions of the present problem.